If having the codex has any effect on Kal-El at all, there's no indication of that in the movie.
Kal-El is Superman because he grew up on Earth, under our yellow sun. Zod and the other Kryptonians become powerful under the light of the sun, but it's not the same as having grown up under its influence. This is the main reason Kal-El is so much stronger.
Beside being implied and generally understood, there are lines in the movie that support this:
- Jor-El believes that his cells will "drink the radiation" from the sun, making him a "god".
Jor-El: Did you find the world?
Lara Lor-Van: We have.
Kelex: Orbiting the main sequence yellow star, just as you said it would.
[Kelex shows them earth]
Jor-El: It’s a young star. His cells will drink it’s radiation. It’s a seemingly intelligent population.
Lara Lor-Van: He’ll be an outcast, a freak. They’ll kill him.
Jor-El: No, he’ll be a God to them.
- Jor-El is surprised when he meets the adult Clark, as he's even stronger than he would've predicted. This is evidence of how growing up on the planet has amplified the effect.
Jor-El: You've grown stronger here than I ever could've imagined. The only way to know how strong, is to keep testing your limits.
- Zod recognizes that Kal-El has adapted to "Earth's ecology". This is contrasted to the Kryptonians, who wear environment suits when they first arrive.
General Zod: You’ve spent a lifetime adapting to earth’s ecology but you never adapted to ours.